Chess Puzzles, a free game by independent developer IggyGames, contains 50 chess puzzles of varying difficulty, starting with (relatively simple) mate in 2 problems, ranging to mate in 5. There are relatively few options -- you start with the simpler puzzles and work yourself up to the more complex ones -- but the format works well for the iPhone, and the implementation is crash-free.
If you get completely stuck, a hint button is there to help. Also, if you think you are on the right track -- but aren't -- your incorrect moves are highlighted by the program. This feature makes it feel like Chess Puzzles is "helping you" become a better player.
Obviously the main drawback is the small number of available games, and the ability to skip to games of higher difficulty. Hopefully the developer adds this in the next update.